Are there more standard patterns besides diamonds which will preserve quad only meshes?
Usually one can catch such topologycal tricks by watching
the vertices (poles) neighbour count. In your example there
is a cube which has one vertex with 5 neighbour and another
with 3. The following example shows one where these poles
are neighbours. As Hount House notet in his comment,
sometimes they are called N-poles and E-poles.



When avoiding triangles, you can reduce or increase edge amount
somewhere by the cost of doing it somewhere else. In your example
on the same side you loose one and another edge. In my example I
added a new edge in the right side, and had to redirect the loop
Flow to the top. When simply removing an edge loop, you take
one edge from one side and one from the front side.

Generall, when using only quads on manifold meshes, every sub-mesh
of it (for example strangly subdivided planes like these) must have
even number of border edges (only one face connected). So a hole
with odd number of edges cannot be filled with quads.

Maybe I am getting too wild now. I am currently
working on a trees retopology, where you can see,
that I mostly only have the usual poles or E-poles.



A similar example to yours (two edges taken
from bottom side), but with another loop Flow:



Must note the tutorial in this related
question. I Recommend to watch all episode.